I am always amazed by the idea of being at the minimum. Because every physical system near the minimum is an harmonic oscillator, and we love harmonic oscillator !
Let us consider a generic potential energy function and suppose that is a stationary point for . A stationary point could be a minimum, maximum or inflection point of a differentiable function, in which the first derivative is null. In other words in a stationary point the function stops to increase or decrease.
If we expand in Taylor series the potential in to the second order we have:
Since is a stationary point for the first derivative is null and the potential becomes:
The first term is a number equal to the value of the potential in the stationaty point. Since a generic potential energy is defined up to an additive constant, we can put . Furthermore, the expression means if we stop the expansion of the potential to the second order we commit an error that is an infinitesima quantity lower than .
Finally, we can express the potential energy near a stationary point in this form:
(where ) that is the potential energy of a harmonic oscillator !
The harmonic oscillator is wherever you look in theoretical Physics (Quantum Mechanics, Quantum Field Theory , Solid State Physics and so on …).
The most important thing is we are able to exactly solve the harmonic oscillator, and then you can solve every physical system near a stationary point.
For example, consider the Lennard – Jones potential, phenomenologically describing the short range interaction between neutral particles:
where is the depth of the potential, is the particles relative distance and is the stationary point. If we expand the LJ potential near the minimum we have:
When the inter-particle distance is approximately equal to they vibrate in harmony !
So, if you stay in the minimum of your potential energy, sin prisa, you can vibrate sin pausa (cit. R.B)